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Metamath Proof Explorer

Theorem elefmndbas

Description: Two ways of saying a function is a mapping of A to itself. (Contributed by AV, 27-Jan-2024)

Ref Expression
Hypotheses efmndbas.g ⊢ G = EndoFMnd ⁡ A
efmndbas.b ⊢ B = Base G
Assertion elefmndbas ⊢ A ∈ V → F ∈ B ↔ F : A ⟶ A

Proof

Step Hyp Ref Expression
1 efmndbas.g ⊢ G = EndoFMnd ⁡ A
2 efmndbas.b ⊢ B = Base G
3 1 2 efmndbas ⊢ B = A A
4 3 eleq2i ⊢ F ∈ B ↔ F ∈ A A
5 id ⊢ A ∈ V → A ∈ V
6 5 5 elmapd ⊢ A ∈ V → F ∈ A A ↔ F : A ⟶ A
7 4 6 bitrid ⊢ A ∈ V → F ∈ B ↔ F : A ⟶ A